Twisting rectangle with text surface
I gave it a try:




Start with a Bezier Circle (Add > Curve > Circle)
Add a cube and change the x-dimension to 2 * pi. You can calculate directly in the input field and you can use constants like pi. It should be 6.28 m and so it fits the circumference.
In Edit Mode gave the cube some additional loop cuts with CTRL+R. Scale the cube to your liking, but keep the x-dimension. In scale-mode press Shift-x to lock scaling on that axis. Delete the end faces. It should look like this: 
Edit: Make even more subdivisions ;)
Add a Simple-Deform-Modifier. Set it's mode to Twist, the Axis to X and the Angle to 360°
Add a Curve-Modifier. Due to our calculation it fits perfectly.
To animate the twisting you can animate the cubes x-rotation
For Texturing unwrap your cube and place your stripes on the text created in an image editor of your choice.
You can animate the texture later. Therefore you can rotate every second stripe around 180° to let them move in opposite directions.
To animate the texture set up your material and animate the x-location.


Hope that helps! ;)